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To carefully graph the exponential function [tex]\( g(x) = 4^x + 3 \)[/tex], we'll take the following steps:
1. Understand the function and its components: The function [tex]\( g(x) = 4^x + 3 \)[/tex] is an exponential function with a base of [tex]\( 4 \)[/tex], vertically shifted upward by [tex]\( 3 \)[/tex]. This means that the graph is the same as [tex]\( 4^x \)[/tex] but moved up 3 units.
2. Determine important points: To plot the graph, it's helpful to calculate the function's values at some key points:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 4^0 + 3 = 1 + 3 = 4 \][/tex]
So, one point is [tex]\((0, 4)\)[/tex].
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = 4^1 + 3 = 4 + 3 = 7 \][/tex]
So, another point is [tex]\((1, 7)\)[/tex].
3. Draw the horizontal asymptote: This function has a horizontal asymptote at [tex]\( y = 3 \)[/tex]. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 4^x \)[/tex] approaches 0, making [tex]\( g(x) \)[/tex] approach 3.
4. Plot the points and asymptote: On graph paper or a graphing tool:
- Plot the two points [tex]\((0, 4)\)[/tex] and [tex]\((1, 7)\)[/tex].
- Draw a dashed horizontal line at [tex]\( y = 3 \)[/tex] to represent the asymptote.
5. Draw the curve: Since [tex]\( 4^x \)[/tex] grows rapidly, draw a smooth curve that starts just above the line [tex]\( y = 3 \)[/tex] for large negative [tex]\( x \)[/tex], passing through the points [tex]\((0, 4)\)[/tex] and [tex]\((1, 7)\)[/tex], and rising steeply for positive [tex]\( x \)[/tex].
6. Determine the domain and range:
- Domain: The function [tex]\( g(x) = 4^x + 3 \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. Hence, the domain is:
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]
- Range: Since [tex]\( 4^x \)[/tex] is positive for all real [tex]\( x \)[/tex] and [tex]\( 4^x \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( g(x) = 4^x + 3 \)[/tex] will always be greater than 3. Thus, the range is:
[tex]\[ \text{Range: } (3, \infty) \][/tex]
### Summary:
- Domain: [tex]\( (-\infty, \infty) \)[/tex]
- Range: [tex]\( (3, \infty) \)[/tex]
With this detailed explanation, you should be able to accurately graph the exponential function [tex]\( g(x) = 4^x + 3 \)[/tex], plot the key points, draw the asymptote, and understand the domain and range of the function.
1. Understand the function and its components: The function [tex]\( g(x) = 4^x + 3 \)[/tex] is an exponential function with a base of [tex]\( 4 \)[/tex], vertically shifted upward by [tex]\( 3 \)[/tex]. This means that the graph is the same as [tex]\( 4^x \)[/tex] but moved up 3 units.
2. Determine important points: To plot the graph, it's helpful to calculate the function's values at some key points:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 4^0 + 3 = 1 + 3 = 4 \][/tex]
So, one point is [tex]\((0, 4)\)[/tex].
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = 4^1 + 3 = 4 + 3 = 7 \][/tex]
So, another point is [tex]\((1, 7)\)[/tex].
3. Draw the horizontal asymptote: This function has a horizontal asymptote at [tex]\( y = 3 \)[/tex]. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 4^x \)[/tex] approaches 0, making [tex]\( g(x) \)[/tex] approach 3.
4. Plot the points and asymptote: On graph paper or a graphing tool:
- Plot the two points [tex]\((0, 4)\)[/tex] and [tex]\((1, 7)\)[/tex].
- Draw a dashed horizontal line at [tex]\( y = 3 \)[/tex] to represent the asymptote.
5. Draw the curve: Since [tex]\( 4^x \)[/tex] grows rapidly, draw a smooth curve that starts just above the line [tex]\( y = 3 \)[/tex] for large negative [tex]\( x \)[/tex], passing through the points [tex]\((0, 4)\)[/tex] and [tex]\((1, 7)\)[/tex], and rising steeply for positive [tex]\( x \)[/tex].
6. Determine the domain and range:
- Domain: The function [tex]\( g(x) = 4^x + 3 \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. Hence, the domain is:
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]
- Range: Since [tex]\( 4^x \)[/tex] is positive for all real [tex]\( x \)[/tex] and [tex]\( 4^x \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( g(x) = 4^x + 3 \)[/tex] will always be greater than 3. Thus, the range is:
[tex]\[ \text{Range: } (3, \infty) \][/tex]
### Summary:
- Domain: [tex]\( (-\infty, \infty) \)[/tex]
- Range: [tex]\( (3, \infty) \)[/tex]
With this detailed explanation, you should be able to accurately graph the exponential function [tex]\( g(x) = 4^x + 3 \)[/tex], plot the key points, draw the asymptote, and understand the domain and range of the function.
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